In the northern Adriatic Sea, the First World War did something odd to the fish market. Fishing boats stayed in port, and when they came back afterwards the catch had changed: predators such as sharks, skates and rays made up a bigger share than before. Fewer nets should have meant more of everything. Why did the hunters gain on the hunted?
The biologist Umberto D'Ancona had the fish records and no explanation. His father-in-law was Vito Volterra, a mathematician, who turned the question into two equations. The American chemist Alfred Lotka had reached similar equations on his own, which is why they carry both names.
The model has only two rules. Prey multiply when left alone. Predators die off when left alone. Every meeting between the two helps the predator and hurts the prey. That is all, and it is enough to make both populations rise and fall in a loop, with the predators always peaking a little after the prey.
Think of rabbits and foxes. Rabbits boom, foxes find food and boom after them, the foxes eat the rabbits down, and then the foxes starve. With the foxes gone the rabbits recover, and the loop begins again.
Drag a point on the phase map. The path it traces is a closed loop. Start near the centre and the loop is small; start near a corner and the swings are huge. Nothing in the model pulls the populations back to a quiet middle, so each starting point keeps its own loop forever.
Now add foxes with the button. The populations do not settle. They jump onto a different loop, often a bigger one with a bigger boom and a deeper crash. Remove half the rabbits and the same thing happens.
Next, raise the fishing slider. Fishing here takes the same share of both species, and the loop's centre moves: the average number of prey goes up, and the average number of predators goes down. Fishing less does the reverse. That is the answer to the Adriatic puzzle. Fishing hits the predators harder in effect, because they only thrive on a healthy supply of prey, while the prey can bounce back. When the nets stayed away, predators gained. This is now called Volterra's principle.
Not quite. In the model nothing ever damps the swings, and populations can dip to a fraction of one animal and still bounce back. Real animals come in whole numbers, so a dip that deep would be extinction. Real food runs out, weather changes and other species join in.
The best-known real example is the snowshoe hare and the Canada lynx, whose numbers rise and fall roughly every ten years. Fur-trade records show it, and lynx follow the hares. But field experiments in the Yukon suggest that predators are only part of the story: the hares' own food supply matters a great deal too. The loop is a good first picture of why populations swing, not the whole machine.
Still, the lesson holds: in a system where each species feeds on the other, doing less can change who wins.